WASHINGTON—In a striking policy shift that ensured the rancorous spy-policy debate would extend into the fall, President Barack Obama announced plans to overhaul key parts the National Security Agency's surveillance programs.

The proposal—an extraordinary step in the face of this year's revelations from fugitive NSA leaker Edward Snowden—drew sharp responses from Republican lawmakers who suggested the president was retreating under political pressure. The tone of the reaction from Republicans and skeptical Democrats...